\(\int \frac {(a+b \sqrt [3]{x})^{10}}{x^8} \, dx\) [2336]

Optimal result

Integrand size = 15, antiderivative size = 144 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^8} \, dx=-\frac {a^{10}}{7 x^7}-\frac {3 a^9 b}{2 x^{20/3}}-\frac {135 a^8 b^2}{19 x^{19/3}}-\frac {20 a^7 b^3}{x^6}-\frac {630 a^6 b^4}{17 x^{17/3}}-\frac {189 a^5 b^5}{4 x^{16/3}}-\frac {42 a^4 b^6}{x^5}-\frac {180 a^3 b^7}{7 x^{14/3}}-\frac {135 a^2 b^8}{13 x^{13/3}}-\frac {5 a b^9}{2 x^4}-\frac {3 b^{10}}{11 x^{11/3}} \]

[Out]

-1/7*a^10/x^7-3/2*a^9*b/x^(20/3)-135/19*a^8*b^2/x^(19/3)-20*a^7*b^3/x^6-630/17*a^6*b^4/x^(17/3)-189/4*a^5*b^5/
x^(16/3)-42*a^4*b^6/x^5-180/7*a^3*b^7/x^(14/3)-135/13*a^2*b^8/x^(13/3)-5/2*a*b^9/x^4-3/11*b^10/x^(11/3)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^8} \, dx=-\frac {a^{10}}{7 x^7}-\frac {3 a^9 b}{2 x^{20/3}}-\frac {135 a^8 b^2}{19 x^{19/3}}-\frac {20 a^7 b^3}{x^6}-\frac {630 a^6 b^4}{17 x^{17/3}}-\frac {189 a^5 b^5}{4 x^{16/3}}-\frac {42 a^4 b^6}{x^5}-\frac {180 a^3 b^7}{7 x^{14/3}}-\frac {135 a^2 b^8}{13 x^{13/3}}-\frac {5 a b^9}{2 x^4}-\frac {3 b^{10}}{11 x^{11/3}} \]

[In]

Int[(a + b*x^(1/3))^10/x^8,x]

[Out]

-1/7*a^10/x^7 - (3*a^9*b)/(2*x^(20/3)) - (135*a^8*b^2)/(19*x^(19/3)) - (20*a^7*b^3)/x^6 - (630*a^6*b^4)/(17*x^
(17/3)) - (189*a^5*b^5)/(4*x^(16/3)) - (42*a^4*b^6)/x^5 - (180*a^3*b^7)/(7*x^(14/3)) - (135*a^2*b^8)/(13*x^(13
/3)) - (5*a*b^9)/(2*x^4) - (3*b^10)/(11*x^(11/3))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {(a+b x)^{10}}{x^{22}} \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int \left (\frac {a^{10}}{x^{22}}+\frac {10 a^9 b}{x^{21}}+\frac {45 a^8 b^2}{x^{20}}+\frac {120 a^7 b^3}{x^{19}}+\frac {210 a^6 b^4}{x^{18}}+\frac {252 a^5 b^5}{x^{17}}+\frac {210 a^4 b^6}{x^{16}}+\frac {120 a^3 b^7}{x^{15}}+\frac {45 a^2 b^8}{x^{14}}+\frac {10 a b^9}{x^{13}}+\frac {b^{10}}{x^{12}}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {a^{10}}{7 x^7}-\frac {3 a^9 b}{2 x^{20/3}}-\frac {135 a^8 b^2}{19 x^{19/3}}-\frac {20 a^7 b^3}{x^6}-\frac {630 a^6 b^4}{17 x^{17/3}}-\frac {189 a^5 b^5}{4 x^{16/3}}-\frac {42 a^4 b^6}{x^5}-\frac {180 a^3 b^7}{7 x^{14/3}}-\frac {135 a^2 b^8}{13 x^{13/3}}-\frac {5 a b^9}{2 x^4}-\frac {3 b^{10}}{11 x^{11/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^8} \, dx=\frac {-184756 a^{10}-1939938 a^9 b \sqrt [3]{x}-9189180 a^8 b^2 x^{2/3}-25865840 a^7 b^3 x-47927880 a^6 b^4 x^{4/3}-61108047 a^5 b^5 x^{5/3}-54318264 a^4 b^6 x^2-33256080 a^3 b^7 x^{7/3}-13430340 a^2 b^8 x^{8/3}-3233230 a b^9 x^3-352716 b^{10} x^{10/3}}{1293292 x^7} \]

[In]

Integrate[(a + b*x^(1/3))^10/x^8,x]

[Out]

(-184756*a^10 - 1939938*a^9*b*x^(1/3) - 9189180*a^8*b^2*x^(2/3) - 25865840*a^7*b^3*x - 47927880*a^6*b^4*x^(4/3
) - 61108047*a^5*b^5*x^(5/3) - 54318264*a^4*b^6*x^2 - 33256080*a^3*b^7*x^(7/3) - 13430340*a^2*b^8*x^(8/3) - 32
33230*a*b^9*x^3 - 352716*b^10*x^(10/3))/(1293292*x^7)

Maple [A] (verified)

Time = 3.49 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(-\frac {a^{10}}{7 x^{7}}-\frac {3 a^{9} b}{2 x^{\frac {20}{3}}}-\frac {135 a^{8} b^{2}}{19 x^{\frac {19}{3}}}-\frac {20 a^{7} b^{3}}{x^{6}}-\frac {630 a^{6} b^{4}}{17 x^{\frac {17}{3}}}-\frac {189 a^{5} b^{5}}{4 x^{\frac {16}{3}}}-\frac {42 a^{4} b^{6}}{x^{5}}-\frac {180 a^{3} b^{7}}{7 x^{\frac {14}{3}}}-\frac {135 a^{2} b^{8}}{13 x^{\frac {13}{3}}}-\frac {5 a \,b^{9}}{2 x^{4}}-\frac {3 b^{10}}{11 x^{\frac {11}{3}}}\)	\(113\)
default	\(-\frac {a^{10}}{7 x^{7}}-\frac {3 a^{9} b}{2 x^{\frac {20}{3}}}-\frac {135 a^{8} b^{2}}{19 x^{\frac {19}{3}}}-\frac {20 a^{7} b^{3}}{x^{6}}-\frac {630 a^{6} b^{4}}{17 x^{\frac {17}{3}}}-\frac {189 a^{5} b^{5}}{4 x^{\frac {16}{3}}}-\frac {42 a^{4} b^{6}}{x^{5}}-\frac {180 a^{3} b^{7}}{7 x^{\frac {14}{3}}}-\frac {135 a^{2} b^{8}}{13 x^{\frac {13}{3}}}-\frac {5 a \,b^{9}}{2 x^{4}}-\frac {3 b^{10}}{11 x^{\frac {11}{3}}}\)	\(113\)
trager	\(\frac {\left (-1+x \right ) \left (2 a^{9} x^{6}+280 a^{6} b^{3} x^{6}+588 x^{6} a^{3} b^{6}+35 b^{9} x^{6}+2 a^{9} x^{5}+280 a^{6} b^{3} x^{5}+588 a^{3} b^{6} x^{5}+35 b^{9} x^{5}+2 a^{9} x^{4}+280 a^{6} b^{3} x^{4}+588 a^{3} b^{6} x^{4}+35 b^{9} x^{4}+2 a^{9} x^{3}+280 a^{6} b^{3} x^{3}+588 a^{3} b^{6} x^{3}+35 b^{9} x^{3}+2 a^{9} x^{2}+280 a^{6} b^{3} x^{2}+588 a^{3} b^{6} x^{2}+2 a^{9} x +280 x \,a^{6} b^{3}+2 a^{9}\right ) a}{14 x^{7}}-\frac {3 \left (238 b^{9} x^{3}+22440 a^{3} b^{6} x^{2}+32340 x \,a^{6} b^{3}+1309 a^{9}\right ) b}{2618 x^{\frac {20}{3}}}-\frac {27 \left (380 b^{6} x^{2}+1729 a^{3} b^{3} x +260 a^{6}\right ) a^{2} b^{2}}{988 x^{\frac {19}{3}}}\)	\(288\)

[In]

int((a+b*x^(1/3))^10/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/7*a^10/x^7-3/2*a^9*b/x^(20/3)-135/19*a^8*b^2/x^(19/3)-20*a^7*b^3/x^6-630/17*a^6*b^4/x^(17/3)-189/4*a^5*b^5/
x^(16/3)-42*a^4*b^6/x^5-180/7*a^3*b^7/x^(14/3)-135/13*a^2*b^8/x^(13/3)-5/2*a*b^9/x^4-3/11*b^10/x^(11/3)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^8} \, dx=-\frac {3233230 \, a b^{9} x^{3} + 54318264 \, a^{4} b^{6} x^{2} + 25865840 \, a^{7} b^{3} x + 184756 \, a^{10} + 35343 \, {\left (380 \, a^{2} b^{8} x^{2} + 1729 \, a^{5} b^{5} x + 260 \, a^{8} b^{2}\right )} x^{\frac {2}{3}} + 1482 \, {\left (238 \, b^{10} x^{3} + 22440 \, a^{3} b^{7} x^{2} + 32340 \, a^{6} b^{4} x + 1309 \, a^{9} b\right )} x^{\frac {1}{3}}}{1293292 \, x^{7}} \]

[In]

integrate((a+b*x^(1/3))^10/x^8,x, algorithm="fricas")

[Out]

-1/1293292*(3233230*a*b^9*x^3 + 54318264*a^4*b^6*x^2 + 25865840*a^7*b^3*x + 184756*a^10 + 35343*(380*a^2*b^8*x
^2 + 1729*a^5*b^5*x + 260*a^8*b^2)*x^(2/3) + 1482*(238*b^10*x^3 + 22440*a^3*b^7*x^2 + 32340*a^6*b^4*x + 1309*a
^9*b)*x^(1/3))/x^7

Sympy [A] (verification not implemented)

Time = 1.09 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^8} \, dx=- \frac {a^{10}}{7 x^{7}} - \frac {3 a^{9} b}{2 x^{\frac {20}{3}}} - \frac {135 a^{8} b^{2}}{19 x^{\frac {19}{3}}} - \frac {20 a^{7} b^{3}}{x^{6}} - \frac {630 a^{6} b^{4}}{17 x^{\frac {17}{3}}} - \frac {189 a^{5} b^{5}}{4 x^{\frac {16}{3}}} - \frac {42 a^{4} b^{6}}{x^{5}} - \frac {180 a^{3} b^{7}}{7 x^{\frac {14}{3}}} - \frac {135 a^{2} b^{8}}{13 x^{\frac {13}{3}}} - \frac {5 a b^{9}}{2 x^{4}} - \frac {3 b^{10}}{11 x^{\frac {11}{3}}} \]

[In]

integrate((a+b*x**(1/3))**10/x**8,x)

[Out]

-a**10/(7*x**7) - 3*a**9*b/(2*x**(20/3)) - 135*a**8*b**2/(19*x**(19/3)) - 20*a**7*b**3/x**6 - 630*a**6*b**4/(1
7*x**(17/3)) - 189*a**5*b**5/(4*x**(16/3)) - 42*a**4*b**6/x**5 - 180*a**3*b**7/(7*x**(14/3)) - 135*a**2*b**8/(
13*x**(13/3)) - 5*a*b**9/(2*x**4) - 3*b**10/(11*x**(11/3))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^8} \, dx=-\frac {352716 \, b^{10} x^{\frac {10}{3}} + 3233230 \, a b^{9} x^{3} + 13430340 \, a^{2} b^{8} x^{\frac {8}{3}} + 33256080 \, a^{3} b^{7} x^{\frac {7}{3}} + 54318264 \, a^{4} b^{6} x^{2} + 61108047 \, a^{5} b^{5} x^{\frac {5}{3}} + 47927880 \, a^{6} b^{4} x^{\frac {4}{3}} + 25865840 \, a^{7} b^{3} x + 9189180 \, a^{8} b^{2} x^{\frac {2}{3}} + 1939938 \, a^{9} b x^{\frac {1}{3}} + 184756 \, a^{10}}{1293292 \, x^{7}} \]

[In]

integrate((a+b*x^(1/3))^10/x^8,x, algorithm="maxima")

[Out]

-1/1293292*(352716*b^10*x^(10/3) + 3233230*a*b^9*x^3 + 13430340*a^2*b^8*x^(8/3) + 33256080*a^3*b^7*x^(7/3) + 5
4318264*a^4*b^6*x^2 + 61108047*a^5*b^5*x^(5/3) + 47927880*a^6*b^4*x^(4/3) + 25865840*a^7*b^3*x + 9189180*a^8*b
^2*x^(2/3) + 1939938*a^9*b*x^(1/3) + 184756*a^10)/x^7

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^8} \, dx=-\frac {352716 \, b^{10} x^{\frac {10}{3}} + 3233230 \, a b^{9} x^{3} + 13430340 \, a^{2} b^{8} x^{\frac {8}{3}} + 33256080 \, a^{3} b^{7} x^{\frac {7}{3}} + 54318264 \, a^{4} b^{6} x^{2} + 61108047 \, a^{5} b^{5} x^{\frac {5}{3}} + 47927880 \, a^{6} b^{4} x^{\frac {4}{3}} + 25865840 \, a^{7} b^{3} x + 9189180 \, a^{8} b^{2} x^{\frac {2}{3}} + 1939938 \, a^{9} b x^{\frac {1}{3}} + 184756 \, a^{10}}{1293292 \, x^{7}} \]

[In]

integrate((a+b*x^(1/3))^10/x^8,x, algorithm="giac")

[Out]

-1/1293292*(352716*b^10*x^(10/3) + 3233230*a*b^9*x^3 + 13430340*a^2*b^8*x^(8/3) + 33256080*a^3*b^7*x^(7/3) + 5
4318264*a^4*b^6*x^2 + 61108047*a^5*b^5*x^(5/3) + 47927880*a^6*b^4*x^(4/3) + 25865840*a^7*b^3*x + 9189180*a^8*b
^2*x^(2/3) + 1939938*a^9*b*x^(1/3) + 184756*a^10)/x^7

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^8} \, dx=-\frac {\frac {a^{10}}{7}+\frac {3\,b^{10}\,x^{10/3}}{11}+20\,a^7\,b^3\,x+\frac {5\,a\,b^9\,x^3}{2}+\frac {3\,a^9\,b\,x^{1/3}}{2}+42\,a^4\,b^6\,x^2+\frac {135\,a^8\,b^2\,x^{2/3}}{19}+\frac {630\,a^6\,b^4\,x^{4/3}}{17}+\frac {189\,a^5\,b^5\,x^{5/3}}{4}+\frac {180\,a^3\,b^7\,x^{7/3}}{7}+\frac {135\,a^2\,b^8\,x^{8/3}}{13}}{x^7} \]

[In]

int((a + b*x^(1/3))^10/x^8,x)

[Out]

-(a^10/7 + (3*b^10*x^(10/3))/11 + 20*a^7*b^3*x + (5*a*b^9*x^3)/2 + (3*a^9*b*x^(1/3))/2 + 42*a^4*b^6*x^2 + (135
*a^8*b^2*x^(2/3))/19 + (630*a^6*b^4*x^(4/3))/17 + (189*a^5*b^5*x^(5/3))/4 + (180*a^3*b^7*x^(7/3))/7 + (135*a^2
*b^8*x^(8/3))/13)/x^7